Hello, boxcarracer!

This is not a simple problem.

. . I hope I got it right . . .

A manufacturer of merry-go-rounds uses 8 identical wooden horses,

4 identical plastic motorbikes and 2 different miniature car.

They are all equally connected around the rim of a circular moving base.

Establish how many different arrangements there can be

if the two cars are not to be placed in consecutive positions.

There are 8 identical horses, 4 identical bikes and 2 different cars.

The objects are: .

If they were arranged in arow, there are: . arrangements.

. . Since they are in a circle, there are: . arrangements.

Now we'll find the number of ways the two carsaretogether.

Duct-tape the two cars together.

Then we have 13 "objects" to arrange: .

. . In arow, there are: . arrangements.

But the two cars could be taped like this: .

. . which creates another 6435 arrangements, for a total of 12,870 ways.

But since they are arranged in a circle, there are: . ways.

Recap: .There are 6435 possible arrangements.

. . . . . .In 990 of them, the two cars are adjacent.

Therefore, there are: . ways in which the cars arenotadjacent.